2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112799We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and discuss the group structure for these transformations.Differential GeometryMathematical Physics53A15, 37K, 22E67text